63edo: Difference between revisions

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== Theory ==

~~The equal temperament~~ [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the 29 & 34d temperament in the 7-, 11- and 13-limit.

63edo [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap|29 & 34d}} temperament in the 7-, 11- and 13-limit.

63 is also a fascinating division to look at in the [[47-limit]]. Although it does not deal as well with primes 5, 17, 19, 37 and 41, it excels in the 2.3.7.11.13.23.29.31.43.47 [[subgroup]], and is a great candidate for a [[gentle]] tuning. Its regular augmented fourth (+6 fifths) is less than 0.3 cents sharp of [[23/16]], therefore tempering out [[736/729]]. Its diesis (+12 fifths) can represent [[33/32]], [[32/31]], [[30/29]], [[29/28]], [[28/27]], as well as [[91/88]], and more, so it is very versatile, making chains of fifths of 12 tones or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. We can take advantage of the representation of 27:28:29:30:31:32:33, which splits [[11/9]] into six "small dieses" as a result; here it can be seen more clearly why these are not regular quarter-tones so are best distinguished from such with the qualifier "large", as otherwise we would expect to see some flavour of minor third after six of them.

A 17-tone fifths chain looks on the surface a little similar to [[17edo]], but as ~~-17~~ fifths gets us to [[64/63]], observing the comma becomes an essential part in progressions favouring prime 7. Furthermore, its prime 5 is far from unusable; although [[25/16]] is barely inconsistent, this affords the tuning supporting 7-limit magic, which may be considered interesting or desirable in of itself. And if this was not enough, if you really want to, it offers reasonable approximations to some yet higher primes too; namely [[43/32]], [[47/32]] and [[53/32]]; see the tables below.

A 17-tone fifths chain looks on the surface a little similar to [[17edo]], but as −17 fifths gets us to [[64/63]], observing the comma becomes an essential part in progressions favouring prime 7. Furthermore, its prime 5 is far from unusable; although [[25/16]] is barely inconsistent, this affords the tuning supporting 7-limit magic, which may be considered interesting or desirable in of itself. And if this was not enough, if you really want to, it offers reasonable approximations to some yet higher primes too; namely [[43/32]], [[47/32]], and [[53/32]]; see the tables below.

=== Prime harmonics ===

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The following table was created using [[User:Godtone#My python 3 code|Godtone's code]] with the command <code><nowiki>interpret_edo(63,ol=47,no=[5,17,19,25,27,37,41],add=[63,73,75,87,89,91,93],dec="''",wiki=23)</nowiki></code> (run in a Python 3 interactive console) plus manual correction of the order of some inconsistent intervals.

As the command indicates, it is a(n accurate) no-5's no-17's no-19's no-25's no-27's no-37's no-41's 47-odd-limit add-63 add-73 add-75 add-87 add-89 add-91 add-93 interpretation, tuned to the strengths of [[63edo]]. Note that because of the cancellation of factors, some odd harmonics of 5 (the more relevant ones) are present, specifically 75/3 = 25, 45/3 = 15 and 45/9 = 5.

As the command indicates, it is a(n accurate) no-5's no-17's no-19's no-25's no-27's no-37's no-41's 47-odd-limit add-63 add-73 add-75 add-87 add-89 add-91 add-93 interpretation, tuned to the strengths of [[63edo]]. Note that because of the cancellation of factors, some odd harmonics of 5 (the more relevant ones) are present, specifically {{nowrap|75/3 {{=}} 25}}, {{nowrap|45/3 {{=}} 15}}, and {{nowrap|45/9 {{=}} 5}}.

Intervals are listed in order of size, so that one can know their relative order at a glance and deem the value of the interpretation for a harmonic context, and [[23-limit]] intervals are highlighted for navigability as [[13-limit]] intervals are more likely to already have pages, and as we are excluding primes 17 and 19, we are only adding prime 23 to the 13-limit.

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! Degree

! Cents

! Approximate Ratios*

! Approximate ratios<ref group="note">{{sg|limit=2.3.5.7.11.13.23.29.31.43.47.73-subgroup (no-17's no-19's no-37's no-41's 47-limit add-73 add-89)}}. Accurate or low-complexity intervals involving 5 are also included here.</ref>

|-

| 0

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| [[2/1]]

|}

~~<nowiki>*</nowiki> as a 2.3.5.7.11.13.23.29.31.43.47.73-subgroup (no-17's no-19's no-37's no-41's 47-limit add-73 add-89) temperament with accurate or low-complexity intervals involving 5 added.~~

== Notation ==

=== Sagittal notation ===

===Sagittal notation===

This notation uses the same sagittal sequence as [[56edo#Sagittal notation|56-EDO]].

~~====Evo flavor====~~

==== Evo flavor ====

File:63-EDO_Evo_Sagittal.svg

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</imagemap>

====Revo flavor====

==== Revo flavor ====

File:63-EDO_Revo_Sagittal.svg

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default [[File:63-EDO_Revo_Sagittal.svg]]

</imagemap>

=== Ups and downs notation ===

Using [[Helmholtz–Ellis]] accidentals, 49edo can be notated using [[ups and downs notation]]:

== Zeta properties ==

===Zeta peak index===

=== Zeta peak index ===

{| class="wikitable"

~~! colspan="3" |Tuning~~

~~! colspan="3" |Strength~~

~~! colspan="2" |Closest EDO~~

~~! colspan="2" |Integer limit~~

|-

!ZPI

! colspan="3" | Tuning

!Steps per octave

! colspan="3" | Strength

!Step size (cents)

! colspan="2" | Closest EDO

!Height

! colspan="2" | Integer limit

!Integral

|-

!Gap

! ZPI

!EDO

! Steps per octave

!Octave (cents)

! Step size (cents)

!Consistent

! Height

!Distinct

! Integral

! Gap

! EDO

! Octave (cents)

! Consistent

! Distinct

|-

|[[321zpi]]

| [[321zpi]]

|63.0192885705350

| 63.0192885705350

|19.0417890652143

| 19.0417890652143

|6.768662

| 6.768662

|1.049023

| 1.049023

|15.412920

| 15.412920

|63edo

| 63edo

|1199.63271110850

| 1199.63271110850

|8

| 8

|8

| 8

|}

== Scales ==

* Approximation of ''[[Pelog]] lima'': 6 9 21 6 21

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* [https://archive.org/details/17_63EDOEarlyDreamsTwo ''early dreams 2''] (2016)

* [https://soundcloud.com/camtaylor-1/63edobosanquetaxis-8thjuly2016-237111323-seconds-and-otonal-shifts ''Seconds and Otonal Shifts''] (2016)

== Notes ==

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 == Theory ==
-The equal temperament [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the 29 &amp; 34d temperament in the 7-, 11- and 13-limit.
+edo [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap|29 &amp; 34d}} temperament in the 7-, 11- and 13-limit.
 is also a fascinating division to look at in the [[47-limit]]. Although it does not deal as well with primes 5, 17, 19, 37 and 41, it excels in the 2.3.7.11.13.23.29.31.43.47 [[subgroup]], and is a great candidate for a [[gentle]] tuning. Its regular augmented fourth (+6 fifths) is less than 0.3 cents sharp of [[23/16]], therefore tempering out [[736/729]]. Its diesis (+12 fifths) can represent [[33/32]], [[32/31]], [[30/29]], [[29/28]], [[28/27]], as well as [[91/88]], and more, so it is very versatile, making chains of fifths of 12 tones or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. We can take advantage of the representation of 27:28:29:30:31:32:33, which splits [[11/9]] into six "small dieses" as a result; here it can be seen more clearly why these are not regular quarter-tones so are best distinguished from such with the qualifier "large", as otherwise we would expect to see some flavour of minor third after six of them.
-A 17-tone fifths chain looks on the surface a little similar to [[17edo]], but as -17 fifths gets us to [[64/63]], observing the comma becomes an essential part in progressions favouring prime 7. Furthermore, its prime 5 is far from unusable; although [[25/16]] is barely inconsistent, this affords the tuning supporting 7-limit magic, which may be considered interesting or desirable in of itself. And if this was not enough, if you really want to, it offers reasonable approximations to some yet higher primes too; namely [[43/32]], [[47/32]] and [[53/32]]; see the tables below.
+A 17-tone fifths chain looks on the surface a little similar to [[17edo]], but as −17 fifths gets us to [[64/63]], observing the comma becomes an essential part in progressions favouring prime 7. Furthermore, its prime 5 is far from unusable; although [[25/16]] is barely inconsistent, this affords the tuning supporting 7-limit magic, which may be considered interesting or desirable in of itself. And if this was not enough, if you really want to, it offers reasonable approximations to some yet higher primes too; namely [[43/32]], [[47/32]], and [[53/32]]; see the tables below.
 === Prime harmonics ===
@@ Line 19: / Line 19: @@
 The following table was created using [[User:Godtone#My python 3 code|Godtone's code]] with the command <code><nowiki>interpret_edo(63,ol=47,no=[5,17,19,25,27,37,41],add=[63,73,75,87,89,91,93],dec="''",wiki=23)</nowiki></code> (run in a Python 3 interactive console) plus manual correction of the order of some inconsistent intervals.
-As the command indicates, it is a(n accurate) no-5's no-17's no-19's no-25's no-27's no-37's no-41's 47-odd-limit add-63 add-73 add-75 add-87 add-89 add-91 add-93 interpretation, tuned to the strengths of [[63edo]]. Note that because of the cancellation of factors, some odd harmonics of 5 (the more relevant ones) are present, specifically 75/3 = 25, 45/3 = 15 and 45/9 = 5.
+As the command indicates, it is a(n accurate) no-5's no-17's no-19's no-25's no-27's no-37's no-41's 47-odd-limit add-63 add-73 add-75 add-87 add-89 add-91 add-93 interpretation, tuned to the strengths of [[63edo]]. Note that because of the cancellation of factors, some odd harmonics of 5 (the more relevant ones) are present, specifically {{nowrap|75/3 {{=}} 25}}, {{nowrap|45/3 {{=}} 15}}, and {{nowrap|45/9 {{=}} 5}}.
 Intervals are listed in order of size, so that one can know their relative order at a glance and deem the value of the interpretation for a harmonic context, and [[23-limit]] intervals are highlighted for navigability as [[13-limit]] intervals are more likely to already have pages, and as we are excluding primes 17 and 19, we are only adding prime 23 to the 13-limit.
@@ Line 29: / Line 29: @@
 ! Degree
 ! Cents
-! Approximate Ratios*
+! Approximate ratios<ref group="note">{{sg|limit=2.3.5.7.11.13.23.29.31.43.47.73-subgroup (no-17's no-19's no-37's no-41's 47-limit add-73 add-89)}}. Accurate or low-complexity intervals involving 5 are also included here.</ref>
 |-
 | 0
@@ Line 287: / Line 287: @@
 | [[2/1]]
 |}
-<nowiki>*</nowiki> as a 2.3.5.7.11.13.23.29.31.43.47.73-subgroup (no-17's no-19's no-37's no-41's 47-limit add-73 add-89) temperament with accurate or low-complexity intervals involving 5 added.
 == Notation ==
+=== Sagittal notation ===
-===Sagittal notation===
 This notation uses the same sagittal sequence as [[56edo#Sagittal notation|56-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:63-EDO_Evo_Sagittal.svg
@@ Line 306: / Line 304: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:63-EDO_Revo_Sagittal.svg
@@ Line 318: / Line 315: @@
 default [[File:63-EDO_Revo_Sagittal.svg]]
 </imagemap>
+=== Ups and downs notation ===
+Using [[Helmholtz–Ellis]] accidentals, 49edo can be notated using [[ups and downs notation]]:
+{{Sharpness-sharp7}}
 == Zeta properties ==
-===Zeta peak index===
+=== Zeta peak index ===
 {| class="wikitable"
-! colspan="3" |Tuning
-! colspan="3" |Strength
-! colspan="2" |Closest EDO
-! colspan="2" |Integer limit
 |-
-!ZPI
+! colspan="3" | Tuning
-!Steps per octave
+! colspan="3" | Strength
-!Step size (cents)
+! colspan="2" | Closest EDO
-!Height
+! colspan="2" | Integer limit
-!Integral
+|-
-!Gap
+! ZPI
-!EDO
+! Steps per octave
-!Octave (cents)
+! Step size (cents)
-!Consistent
+! Height
-!Distinct
+! Integral
+! Gap
+! EDO
+! Octave (cents)
+! Consistent
+! Distinct
 |-
-|[[321zpi]]
+| [[321zpi]]
-|63.0192885705350
+| 63.0192885705350
-|19.0417890652143
+| 19.0417890652143
-|6.768662
+| 6.768662
-|1.049023
+| 1.049023
-|15.412920
+| 15.412920
-|63edo
+| 63edo
-|1199.63271110850
+| 1199.63271110850
-|8
+| 8
-|8
+| 8
 |}
 == Scales ==
 * Approximation of ''[[Pelog]] lima'': 6 9 21 6 21
@@ Line 360: / Line 363: @@
 * [https://archive.org/details/17_63EDOEarlyDreamsTwo ''early dreams 2''] (2016)
 * [https://soundcloud.com/camtaylor-1/63edobosanquetaxis-8thjuly2016-237111323-seconds-and-otonal-shifts ''Seconds and Otonal Shifts''] (2016)
+== Notes ==
+<references group="note" />
 [[Category:Listen]]